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I am looking for a model, characterized continuously by a single parameter, to describe the arrival times of buses with unit expected interarrival time. At one extreme of the parameter (say $\theta=1$), the process is deterministic, with all interarrival times equal (to $1$). When $\theta=0$, the process is pure Poisson (with unit expected interarrival or waiting time). As $\theta$ approaches $-1$, the arrivals tend to cluster, with long intervals between the clusters. At the (unattainable) extreme of $\theta=-1$, all the buses arrive in a simultaneous convoy, after an infinite wait, and you have to wait forever for the next convoy.

The choice of $\{1,0,-1\}$ for the extreme and central parameter values isn't important: they could be $\{-\infty, 0, \infty\}$, $\{0,\frac12,1\}$, ..., whatever. At this stage, simplicity and naturalness, subject to the above conditions, is more important than realism.

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In a pure Poisson process, the interarrival times (times between each bus) has an exponential distribution. So you want some family of distributions which includes the exponential as a special case. The gamma distribution (see: https://en.wikipedia.org/wiki/Gamma_distribution). When then shape parameter is 1, you have the exponential. When the shape parameter ($k$ or $\alpha$ in wiki) approaches zero, you are approaching constant interarrival times, and growing above 1 you get interarrival times more variable than for the poisson process.

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  • $\begingroup$ Yes, this could be the model if we set $k\theta=1$. But I think that you have it the wrong way round: We approach a constant (unit) interarrival time as $k\rightarrow\infty$, and long intervals between clusters as $k\rightarrow0$. $\endgroup$ – John Bentin Feb 12 '15 at 16:06

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